The partials (component notes of a timbre)
of a string instrument are very loud. In a cello concerto, the
partials will at times be louder than many instruments of the
orchestra, though they are usually not heard as separate notes
unless one listens out for them with a keen ear.

In this clip cello partials, try listening to hear how the note played on the
pan pipes continues into the cello note that follows it (like a
kind of continuing resonance after the note stops sounding). You
can also try singing the note, and in fact it a nice way to get
into playing just intonation is to play a sustained drone note
on, say, a cello (or on the cello voice of your synth or
soundcard) and then sing the partials in turn, or sing tunes
going up and down the harmonic series.

The just intonation major third may sound
flat to a classically trained western musician when first heard.
On the other hand, to a classically trained Indian musician, the
major third of classical Western music may sound extremely sharp
on first hearing. Perhaps after singing along with the cello
drone you may understand why - Indian music uses drones a lot.

The seventh harmonic may sound very flat
to a classically trained western musician on first hearing. It
too is used in many types of music (not in Indian music). For
instance, it is sometimes used in Blues / Jazz. It is a seven
limit note, so called because it is divisible by 7, rather than
powers of 2, 3 and 5 as in the numbers used for the five limit
just intonation scale. It is used in the seven limit dominant
seventh. This gives a wonderfully consonant chord, for those who
have the taste for it.

There are two commonly used five limit
versions of this note - 9/5 or 16/9, which give the notes used
for the two common just intonation five limit dominant sevenths.

Here is a clip of the three dominant
sevenths to compare: (each time resolving to the just intonation
major third triad).

One could think of this score as a kind
of tablature for a harp - a harp which can be tuned to any
pattern of pitches one likes. Then each note will correspond to a
string.

A harpist could read this score exactly
as it is, just as she would play any other harp piece. If playing
a harp retuned in that way, it would sound with the new pitches.
When the harpist plays the G'' string, it will sound the 1/4
below the c string, and so play a C' instead of a G''.

One can also retune ones midi keyboard to
the scale in this same way, using a midi relay program such as my
Fractal Tune Smithy. If one does that, one can play from the
score just as one would from any other score. Again, all the keys
will be retuned accordingly, so that the key which usually plays
c' will instead play the 15/8 above c , and the key which usually
plays the G'' will instead play a C', etc.

So basically, all these scores are
playing scores - retune the midi keyboard, then play the score
exactly as it is written, and because of the way the notes are
retuned, it will sound as intended.

Quite useful also for following along
too, and I find when composing that one soon gets used to the new
layout of the pitches.

Scores are for printing out (or viewing
on very high resolution screen, e.g. 1600 pixels wide).

The dynamics / tempi are more precise
than they need to be - every detail has to be spelt out for the
midi clip. So, if you want to play them, or make mp3 realisations
of them, feel free to be flexible with timing and dynamics.

I used NoteWorthy Composer, which is a
great program, very popular with amateur composers, and I think
deserving the attention of the professionally trained ones too.

The NoteWorthy Composer files can also be
played through midi relaying software retuned according to the
scale shown in the lyric line. I use Fractal Tune Smithy; any midi retuning program can be used.

The one particularly devoted to practical
microtonality is: Crazy music.
Because of a regular poster to Crazy music who,
though an interesting writer on microtonality, often uses
language and metaphors that are offensive to some, a new
moderated group has recently been started called MakeMicroMusic. Crazy music
continues unmoderated. However it is prob. not suitable for young
children ???!! :-)

Jacob Van Eycks Boffons + accompaniment as
exercise for just intonation major third

This is a piece by Jacob Van Eyck, the
dutch recorder player, bell tuner, and carillion player from the
17th century. He is a firm favourite with recorder players
because of his wonderful pieces for the recorder that fit the
instrument so well. Also famous in the history of bell making for
his part in the development of the modern church bell timbre. At
his time he was much famed for his virtuoso playing, which he
played to entertain passers by in the churchyard - people came
from far afield to hear him.

This piece is based around a repeating
sequence of major chords: I, IV, I, V, I, IV, V, I, in the key of
G.

The score is actually shown in the key of
C, with accidentals for the F# where it occurs. Might be a bit of
an anachronism to call it in G major, but that will give an idea.
Has a fair number of F naturals in scale passages, with F
sharps for leading tone type notes and the major thirds of the II
chord.

As a result, it is a wonderful exercise
in playing the just intonation major third.

To make it easier, I've added an
accompaniment as a series of just major chords. Just for fun I've
also added a percussion track in 3/4 played against the 4/4 of
the melody. Not the usual 3 against 4, but a part in 3/4 just
going its own merry way ignoring the 4/4 bar lines, i.e. 3 bars
against 4 (which won't make it easier of course, but fun). That
is just for starters, - the percussion gradually gets totally
zany and crazy and prob. humanly pretty unplayable, but sort of
thing that is easy in a midi file.

Whenever B, E, and F# are played as major
thirds, which is most of the time, they need to be flatter than
normal to be in tune with the accompaniment. On the recorder,
this can be done by a technique that seems to involve varying the
amount of turbulence in the breath (see the Recordings page), in which case, they are nearly as flat as
you can get them while keeping the volume steady. One can also
use finger shading techniques.

Obviously suitable for other instruments
as well.

One needs to have I, IV and V pure in key
of G, so that corresponds to I,V and II in key of C, so one can
use this just intonation scale for the piece:

135/128 9/8 6/5 5/4 27/20 45/32 3/2 8/5
27/16 9/5 15/8 2

One can't have all of I,V, IV and II pure
in G in just intonation, but one can have I, V and II pure if
there are no IVs.

Hexany phrase transformations

If you look at the original piece Hexany phrase, you'll see that I say that this may be seed for
larger piece later on.

Well, here it is. Gene Ward Smith has
developed a technique for transforming the tuning of a tune while
keeping the melodic line intact.

When applied to the hexany, it gives 48
variations.

One can change major chords to a minor
chords and vice versa by inverting everything. Well in the
hexany, there are four types of "major chord", each
with its minor chord inversion. This technique transforms all
these types of major and minor chords into each other in all
possible ways. Some transformations invert the melody, others
leave it the same way up, but transformed into a different type
of major chord, or whatever.

I've made some changes to the original
phrase - completed some of the chords as triads, added a bit more
counterpoint (which is more interesting when the phrase gets
turned upside down), and made sure that all the harmonies are
triads of the hexany. The original phrase has some impure
harmonies that sound quite nice for unresolved chords in the
hexany. These also could be interesting transformed (e.g. it can
be nice to play two triads in the hexany together, with an edge
shared), but for this first try out, seemed best to keep to the
pure triads.

You can transform a tune in this way by
making a midi_remap file with same extension as the midi clip.
Then retune in FTS using a just intonation scale such as the
hexany, (if twelve tone scale, you can try playing simple
diatonic tunes in the just intonation scale, but will prob. need
to use root control to get the chords right before transforming,
for the II chords at least).

The midi remap file needs to have same
name as the source midi clip, with extension .midi_remap (or alternatively, you can make a file called midi_remap - no extension - to transform any midi clips in
the same folder when played by FTS).

So to transform the faster version, here
is the same file as before, but saved under same name as the
source midi clip for the faster version, with extension .midi_remap.

7 limit adaptive puzzle

The soprano and mezzo soprano parts split
into chords occasionally (three soprano parts and two mezzo
sopranos - i.e. like a small choir with the sopranos splitting
into separate voices at that point).

This is a piece that would puzzle a real-time
adaptive retuning program. It is exploring various shades of
sharps and flats.

It starts off in C major, then suddenly
jumps to G# / Ab without any preparation. A series of chords
follow leading back to C. Only when that sequence is finished can
one work out what the original exact pitch of the G# or Ab should
be if one is to keep to the purest possible chords, and end back
at the original pitch for the C. In fact, first time it was an
F#, but at

This then happens a second time, with
another sequence of chords.

In fact, the first time, the note is an
F# at 49/32, in other words, two 7/4s above the C at 1/1. The
second time it is an Ab at 49/32, as two wide 8/7 whole tones
below 1/1 (at least, I think that is one way to interpret it). So
in this one, first time it is an F#, second time it is an Ab, but
both times at the same pitch!

A real time adaptive tuning program
couldn't know in advance if it was going to be a 49/32 or an 8/5
(the more usual ratio for an Ab), or some of the other
possiblities for F# or Ab. The difference in pitch between 49/32
and 8/5 is 76.0345 cents, or three quarters of a semitone.

If the program makes the wrong decision,
the pitch will drift, and not just slighlty, but by three
quarters of a semitone each time it does it. (Or alternatively,
it will have to make some rapid adjustments of pitch of notes
after they are played, which prob. won't sound that good). Since
this unprepared Ab / G# occurs three times in all, the program
could end up over a tone sharp at the end of the tune, if it
chooses to play them all as 8/5s!

It's 7 limit, which means it uses seventh
harmonic notes (such as 7/4 in ratio notation). These tunings are
used in jazz / blues, and occur in many scales from around the
world, but are rather rare in Western classical music as normally
played / sung.

The root control voice shows which note
to use as the root of the scale. E.g. if it shows a C then going
up by semitones from C will give David Canright's 7 limit twelve
tone scale. If it shows an F#, then going up by semitones from F#
will give this scale, and so on.

Here is a log of all the intervals
played, including all intervals between pairs of notes:

Plain ratios are the notes played, and
the ratios between ~s are the intervals of the chords. Can have
two or more ratios, e.g. before top note of a triad, shows the
intervals between that note and both the lower notes.

E.g.:

49/48~3/2~49/32~12/7,
8/7~7/4

A leisure time adaptive retuning program
has a better chance of making a sensible retuning of it.

Soft vertical springs let the intervals
of the chords vary while keeping notes comparatively steady in
pitch throughout the piece. Rigid vertical springs work the other
way - notes vary in pitch, while intervals of chords remain
steady.

Here is the original 12-tet, just to show how much a difference the adaptive
retuning makes. This piece was originally written in 7-limit
rather than 12-tet, and I think it doesn't work too well in 12-tet,
especially at the point when all the black notes are sounded
together to make a chord.

This is a piece that would puzzle a real-time
adaptive retuning program. It has three versions of the Ab / G#,
and the only way to figure out which is to be which is to look
several bars ahead (see below).

Starts with pentatonic / diatonic melody
in C major. The melody is somewhat inspired by the modal feel of
old folk melodies, and the melody could be thought of as in A
minor / mixolydian equally well.

Then you suddenly hear a high Ab on the
violin. Or is it a G#?

The other instruments join in, and after
a while it resolves down in the progression G# (8/9) F# (5/4) D (4/3)
G (4/3) C.

So in fact it was a G# at 3^4*5/2^7 (405/128).

(if you look at the score, then at that
point the root was F#, which in the twelve tone j.i. scale used
is 45/32, and it is 9/8 above that).

Then the pentatonic / diatonic melody
repeats, then you hear the high Ab /G# again. This time it
resolves in progression A# E C. So it was a 25/16 this time (two
major thirds up from the 1/1).

While resolving to the C, it goes down to
Ab for a moment to take in a Ab major chord, so we get the 8/5 Ab
here as well. Returns to C with a couple of bars in C minor.

Back to the original melody again, then
for coda, you hear the high Ab / G# again. If you look at the
score, the root at this point is E, so it is the 25/16 G# again.
This time the key signature is E major, so the melody gets
changed a little, starting from the third degree of the major
scale rather than the second (the A# becomes an A natural). It
starts resolving back via the E as before. However on the way
back, it changes direction twice in quick succession via a couple
of diesis shifts, pure minor thirds changing in pitch as they are
played. These are highly audible (and meant to be) as melodic
shifts with small steps in the melody.

The new section plays around with the
tunes (a phrase from each), in A minor / mixolydian, to bring out
this aspect of the original melody. It then ends with the
original tune once more, but on this last return, it ends on a
single note A, which leaves the whole piece ambiguous between A
minor and C major.

Here is a text log of all the notes and
intervals played (apart from the new section):

The plain ratios are the notes, and the
ratios between ~s are the intervals of the chords.

So for example:

(24)
5/6 ~6/5~ 1/1 ~5/4~ 5/4 ~2/1~ 5/2
is the A minor chord that opens bar 24, with the 5/6 on the
'cello, the 1/1 and 5/4 on second violin, and the 5/2 on the
first violin.

(I've added the bar numbers in brackets
before each bar).

The puzzle for an adaptive retuning program

The adaptive tunign challenge is to find
the optimal solution that has no overall shift of pitch (and no
fudging of the pitches), which a real-time adaptive tuning
program couldn't possibly do without looking ahead several bars (i.e.
so that you play it with a delay of several bars before the notes
sound). A leisure time adaptive tunign program can look ahead no
problem, so could do it in principle, but I wonder if it might
find it somewhat of a puzzle too?

This puzzle needn't have a unique
solution. It would depend on which note one was keeping steady -
in this piece the C as the 1/1 is same pitch at the end as at the
beginning, but one could choose another note for this. Also some
of the chords could be tuned in various ways depending on which
pairs of notes one wanted to have as low ratio just intonation.

Answer: John de Laubenfels leisure time
adaptive tuning program can cope with it (apart from the diesis
shifts of course). Result sounds sweet too, but interestingly, it
comes up with another solution.

A comma pump is a sequence of chords that
can only be tuned to pure intervals if the pitch of the melody
drifts from beginning to end of the note. E.g. might start at C =
1/1 and end up at C = 80/81.

It's twelve tone in sense of having
twelve notes, but with a huge range of step sizes from 27.2 cents
(a bit over an eighth of a tone) to 203.9 cents (over a tone).
The smallest step is from G to Gb. This piece doesn't actually
use the tiniest step from one note to the next anywhere, but you
can find it in the Kalimba accompaniment for successive off beat
quavers in the triplet that brings one back round to the start of
the repeat, and in that pattern it can perhaps be heard as a kind
of melodic step, passing by very fast.

The stretched octaves make the fifths
purer, though they aren't stretched as much as would be needed to
make them completely pure all the way, (which would make the
octave repeat at 1228.42 cents); instead, the double octave is
about 2428 cents, and the scale is interestingly uneven with some
of the fifths purer than others.

7 equal trio

Dynamic range for playback of midi clips
seems to vary a fair amount depending what you play them on, and
the quiet voices are too quiet when they are played in Quicktime
(commonly used midi player in web browsers). Actually, I think
there is supposed to be a standard for the decibel range of a
midi clip, but if so, it doesn't seem to work or be adhered to
too closely.

Notes

Instruments: Violin, viola and
glockenspiel, joined by cello for second movement.

First movement is in 3/8, has a
rather independent glockenspiel part in the middle following its
own way with waves of sound, gradually getting louder and louder
a bit like the sea, which the other instruments basically ignore,
but it sounds okay somehow.

They are joined for the second movement (in
7/4) by a cellist who clearly is a very individual character, and
takes a little while to get into the spirit of things. However,
has a very interesting idea to contribute, which is repeated over
and over, and the others eventually take up on it.

Third movement is a very conventional
seeming 4/4 somewhat after style of Haydn. You'd hardly think
that the parallel thirds are actually 11/9s (pretty close) and
the cadences are III to I (11/9 taking place of 3/2) rather than
V to I.

Last movement is in a lyrical 11/4.

About the scale:

7 tone equal temperament (7-tet) has
seven equally spaced notes to the octave. Near seven equal scales
are traditional in Thailand and Mozambique (see the Chopi scale
on the improvisations page).

7-tet is an interesting scale to compose
in because it turns all ones expectations on their head.

The fifth is a dissonance with strong
beating, as is the fourth. The consonances in this scale are the
third, and sixth.

The third is actually a neutral third,
halfway between the pure minor third at 6/5 and the pure major
third at 5/4. In fact it is almost exactly equal to 11/9, which
is why it sounds so nice - 11/9 is a pleasant diad.

I find that in 7-tet one uses a kind of
cadence involving movement by 11/9 where normally one would use
movment by 3/2. I.e. III to I instead of V to I.

The 11/9 diad is a more complex interval
than a 6/5 or 5/4, and I find it is interesting enough to the ear
to take the place of a triad in cadences.

The triads in 7-tet are very harsh
sounding, if one thinks of them as triads, because of the
dissonant fifth with the relatively pure third. However, when you
add a fourth note on the top you get a nice kind of diminished
seventh type chord, with neutral thirds for the intervals instead
of minor thirds. I think of the 7-tet triads as a kind of
incomplete neutral diminished seventh.

7-tet is a very easy scale to compose in
because all the steps are the same size. So, if you transpose a
melodic phrase up or down by one step, or two steps (as for the
cadences), or whatever, it will be identical, and then you can
repeat that transposition as often as you like. It's not so easy
to improvise in I find! However you'll find an improvisation in
it on the improvisations page.

Quarter comma meantone has pure major
thirds, and reasonable fifths, but one major third in every three
is sharp, and one of the twelve fifths is a wolf fifth, which is
very sharp and unplayable.

It is usually tuned so that the wolf
fifth is in a remote key, so that it is seldom used. Also tuned
so that scales like C major and A major have only pure major
thirds. At the time, it was thought a reasonalbe trade off, to
have one unplayable scale, in order to have pure major thirds in
most of the other scales. For this piece, I chose the wolf fifth
between D# and Bb.

A bit of history: at the time of J.S.
Bach, quarter comma meantone was still a very popular scale, but
used less often than before (though still a very prevalent tuning
for church organs for long after). Other scales were used that
let one modulate more freely to remote scales, such as
Werckmeister III; these have somewhat sharper major thirds,
closer to the ones we are used to today.

The modern twelve tone equal temperament
was used regularly for lutes as it was fairly easy to lay out the
frets for it, and it sounds good on a lute. See Margo Schulter's
post 25000 to the Tuning List.

However, it was not usually used for
keyboards (and doesn't sound so good on a harpsichord because of
the prominent fifth partial in the harpsichord timbre), and
Bach's well tempered clavier was probably intended for a well
tempered scale such as perhaps WIII. Twelve tone equal
temperament may be an early development as one can calculate 2^(1/12)
as the cube root of the square root of the square root of 2. (Alternatively,
one can use the method for finding a^b using logarithms, but this
was only available at end of C17).

Scale is 17-tet major pentatonic. Here 17-tet
stands for seventeen tone equal temperament - in other words you
have seventeen equally spaced notes to an octave. The scale most
pianos are tuned to nowadays, and for most of the last century,
has twelve equally spaced notes to an octave. Before that, pianos
were tuned in a variety of ways. The 17 notes are C, C#, Db, D,
D#, Eb, E, F, F#, Gb, G, G#, Ab, A, A#, Bb, B, all equally spaced.

This is an ordinary pentatonic scale, but
has a very sharp major third, so that one gets a lot of fast
beating if one uses that. Even more if one adds in the fifth as
well. For this reason, somewhat rare to use it, but it has
beautiful interval steps, and depending how it is used, the
beating can be rather attractive.

Notes are C D E G A in 17-tet.

The drone is E A d, or if one uses the
notation with notes numbered by the octave, so that E4 is an
octave above E3, it is E3 A3 D4.

Here I use a drone in consecutive fourths.

There's only one major third in the major
pentatonic, between the C and the E. Listen out for the beating
whenever the melody plays a C, for instance, in the sustained C
of second half of third bar. The beating is fast, so may well
sound just like a somewhat rough or slightly sour note, which may
appeal if one likes the rough sound of some folk instruments and
the hurdy gurdy.

The melody starts and ends on A, and A
minor is the home key of the piece, so I suppose it is really in
the minor pentatonic scale on A.

"I could explain here that the
seventeen-tone system turns certain common rules of harmony
upside-down: major thirds are dissonances which resolve into
fourths instead of the other way round: certain other intervals
resolve into major seconds; the pentatonic scale takes on a very
exciting mood when mapped onto the 17 equally-spaced tones, and
so on; but I can't expect you to believe me until you hear all
this yourself. If you try to play these pieces in another system,
it just doesn't work; they lose their punch; the magic is all
gone. "

Found it while following up a link from
the Huygens-Fokker Foundation biography of him:

7 Limit octony lullaby

String Quartet

This is a string quartet minature in 12-tet
- i.e. twelve equally spaced notes per octave, the standard piano
tuning.

Because it is in 12-tet, there's a fair
amount of beating / roughness of intervals, however we have
learnt to tolerate that, and even to think it sounds nice, and I
like this particular piece tuned this way. Doesn't mean however
that it can only be played this way - you are welcome to try
other tunings too!